论文标题
分层介质中二阶椭圆运算符的本征函数的浓度和不集中
Concentration and non-concentration of eigenfunctions of second-order elliptic operators in layered media
论文作者
论文摘要
这项工作与A = -C $δ$的运营商有关，在域中作用于$ω$：= $ω$'x（0，h）$ \ subseteq $ r ^d x r ^+。扩散系数C> 0取决于$（0，h）中的一个坐标y $ \，但可能是不连续的。这对应于“分层培养基”的物理模型，以声学，弹性，光纤纤维出现... dirichlet边界条件。通常，对于每个$ε$> 0，特征函数的集合分为三个子集的不相交联合：FNG（非引导），FG（指导）和FRES（残留）。残留集的收缩为$ε$ $ \ rightarrow $ 0。在数学文献中分别通过集中/不集中解决方案来代替有指导/非引导的习惯物理术语。对于指导波，“分层介质”的假设使我们能够从集中区域获得严格的指数衰减估计。在文献中，非引导波的案例吸引了较少的关注。尽管它与物理模型没有紧密相关，但它导致了一些有关振荡溶液及其渐近特性的非常有趣的问题。经典的渐近方法可用于$ c 2中的C（y）$ \，但规律性较低，不包括这种方法。相关的本本功能（在FNG中）是振荡的。但是，这一事实本身并不排除两个连续零之间解决方案“扁平”的可能性，从而导致互补段的集中度。在这里，我们表明，当C（y）处于有限的变化时，它不会发生这种情况，这是通过证明“最小振幅假设”的。但是，当c（y）不存在有限变化（即使是连续）时，这种结果的有效性仍然是一个开放的问题。
This work is concerned with operators of the type A = --c$Δ$ acting in domains $Ω$ := $Ω$ ' x (0, H) $\subseteq$ R^d x R ^+. The diffusion coefficient c > 0 depends on one coordinate y $\in$ (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of ''layered media'', appearing in acoustics, elasticity, optical fibers... Dirichlet boundary conditions are assumed. In general, for each $ε$ > 0, the set of eigenfunctions is divided into a disjoint union of three subsets : Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as $ε$ $\rightarrow$ 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of ''layered media'' enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y) $\in$ C 2 but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in Fng) are oscillatory. However, this fact by itself does not exclude the possibility of ''flattening out'' of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a ''minimal amplitude hypothesis''. However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem.
